# Definition:Preimage/Relation/Relation

## Definition

Let $\mathcal R \subseteq S \times T$ be a relation.

Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse relation to $\mathcal R$, defined as:

$\mathcal R^{-1} = \set {\tuple {t, s}: \tuple {s, t} \in \mathcal R}$

The preimage of $\mathcal R \subseteq S \times T$ is:

$\Preimg {\mathcal R} := \mathcal R^{-1} \sqbrk T = \set {s \in S: \exists t \in T: \tuple {s, t} \in \mathcal R}$

## Also known as

Some sources, for example 1975: T.S. Blyth: Set Theory and Abstract Algebra, call this the domain of $\mathcal R$.

However, this term is discouraged, as it is also seen used to mean the entire set $S$, including elements of that set which have no images.

## Technical Note

The $\LaTeX$ code for $\Preimg {f}$ is \Preimg {f} .

When the argument is a single character, it is usual to omit the braces:

\Preimg f